Suppose I have a function that has transformed a sequence of integers like so:
$$ f(n) = \log_{10}\frac{1}{n} $$
I then want to convert those numbers back to $n$. I understand that I first must take $10^{1/n}$ to get rid of my $\log_{10}$ transformation. But how do I go from $1/n$ to just $n$?
For example, let $n = 2$. Then
$$ f(2) = \log_{10}\frac{1}{2} = -0.30103 $$
I can then apply an exponent such that
$$ 10^{-0.30103} = \ \sim.5 $$
What do I need to do to transform $.5$ to $2$? Obviously I can multiply $.5$ by $4$, but I need to be able to do this for any given $n$.
I feel like this is something simple I learned in 8th grade, but I'm having a major relapse of memory at the moment.
$$y = \log_{10}\frac1n$$
$$10^y = \frac1n$$
$$n = \frac1{10^y}=10^{-y}$$
We have $\frac1{1/2}=2$ in particular.